If y = cos - 1 x , then the value of d 2 y d x 2 in terms of y alone is:

Question

If $y = \cos^{- 1} x$ , then the value of $\frac{d^{2} y}{d x^{2}}$ in terms of $y$ alone is:

Smart Achievers · Accepted Answer

(1)

Ans. $- c o t y c o s e c^{2} y$

Explanation:

Given, $y = \cos^{- 1} x$

$\Rightarrow$ $x = \cos y$

On differentiating with respect to $y$ , we get

$\frac{d x}{d y} = - \sin y$

$\Rightarrow$ $\frac{d y}{d x} = - cosec y$ $\dots (i)$

Again, differentiating with respect to $x$ , we get

$\frac{d^{2} y}{d x^{2}} = \frac{d}{d x} (- cosec y) = - (- cosec y \cot y) \frac{d y}{d x}$

$= cosec y \cot y (- cosec y)$

$= - \cot y {cosec}^{2} y$ $[from Eq. (i)]$

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